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Convergence in the mean and almost everywhere of Fourier series in polynomials orthogonal on an interval
V. M. Badkov
Abstract:
Let
$\sigma_p=\{p_n(t)\}_{n=0}^\infty$ be the system of polynomials orthonormal on
$[-1,1]$ with weight
$$
p(t)=H(t)(1-t)^\alpha(1+t)^\beta\prod_{\nu=1}^m|t-x_\nu|^{\gamma_\nu},
$$
where
$-1<x_1<\dots<x_m<1$,
$\alpha,\beta,\gamma_\nu>-1$ (
$\nu=1,\dots,m$),
$H(t)>0$ on
$[-1,1]$ and
$\omega(H,\delta)\delta^{-1}\in L(0,2)$ (
$\omega(H,\delta)$ is the modulus of continuity in
$C(-1,\,1)$). Consider the class of functions
$(qL)^r=\{f(t):q(t)f(t)\in L^r(-1,1)\}$, where $q(t)=(1-t)^A(1+t)^B\times\prod_{\nu=1}^m|t-x_\nu|^{\Gamma_\nu}.$ Let
$S_n^{(p)}(f)=S_n^{(p)}(f,x)$ (
$n=0,1,\dots$) denote the partial sums of the Fourier series of a function
$f$ with repect to the system
$\sigma_p$.
In the paper, conditions are obtained on the exponents of the functions
$p(t)$ and
$q(t)$ and the exponent
$r\in(1,\infty)$ that are necessary and sufficient for the boundedness in
$(qL)^r$ of each of the operators
$S_n^{(p)}(f,x)$ and
$\sup_{n\geqslant0}\{|S_n^{(p)}(f,x)|\}$. Sufficient conditions for the convergence of the partial sums
$S_n^{(p)}(f)$ to
$f\in(qL)^r$ in the mean and almost everywhere in
$(-1,\,1)$ are revealed as a consequence. It is proved that these conditions are best possible on the class
$(qL)^r$ (for
$\omega(H,\delta)\delta^{-1}\in L^2(0,2)$ in the case of convergence almost everywhere). Estimates of the polynomials
$p_n(t)$ and necessary and sufficient conditions for their boundedness in the mean are also obtained.
Bibliography: 26 titles.
UDC:
517.512.7
MSC: 42A20,
42A56 Received: 30.07.1973